Considérons un graphe G = (V,E,w) non orienté dont les sommets sont pondérés et un entier k. Le problème à étudier consiste à la construction des algorithmes afin de déterminer le nombre minimum de nœuds qu’il faut enlever au graphe G pour que toutes les composantes connexes restantes contiennent chacune au plus k-sommets. Ce problème nous l’appelons problème de k-Séparateur et on désigne par k-séparateur le sous-ensemble recherché. Il est une généralisation du Vertex Cover qui correspond au cas k = 1 (nombre minimum de sommets intersectant toutes les arêtes du graphe) / Let G be a vertex-weighted undirected graph. We aim to compute a minimum weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to a given positive number k. If k = 1 we get the classical vertex cover problem. Many formulations are proposed for the problem. The linear relaxations of these formulations are theoretically compared. A polyhedral study is proposed (valid inequalities, facets, separation algorithms). It is shown that the problem can be solved in polynomial time for many special cases including the path, the cycle and the tree cases and also for graphs not containing some special induced sub-graphs. Some (k + 1)-approximation algorithms are also exhibited. Most of the algorithms are implemented and compared. The k-separator problem has many applications. If vertex weights are equal to 1, the size of a minimum k-separator can be used to evaluate the robustness of a graph or a network. Another application consists in partitioning a graph/network into different sub-graphs with respect to different criteria. For example, in the context of social networks, many approaches are proposed to detect communities. By solving a minimum k-separator problem, we get different connected components that may represent communities. The k-separator vertices represent persons making connections between communities. The k-separator problem can then be seen as a special partitioning/clustering graph problem
| Identifer | oai:union.ndltd.org:theses.fr/2014TELE0032 |
| Date | 04 December 2014 |
| Creators | Mohamed Sidi, Mohamed Ahmed |
| Contributors | Evry, Institut national des télécommunications, Ben Ameur, Walid |
| Source Sets | Dépôt national des thèses électroniques françaises |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text |
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